A key question in population biology is understanding the conditions under which the species of an ecosystem persist or go extinct. Theoretical and empirical studies have shown that persistence can be facilitated or negated by both biotic interactions and environmental fluctuations. We study the dynamics of n interacting species that live in a stochastic environment. Our models are described by n dimensional piecewise deterministic Markov processes. These are processes (X(t), r(t)) where the vector X denotes the density of the n species and r(t) is a finite state space process which keeps track of the environment. In any fixed environment the process follows the flow given by a system of ordinary differential equations. The randomness comes from the changes or switches in the environment, which happen at random times. We give sharp conditions under which the populations persist as well as conditions under which some populations go extinct exponentially fast. As an example we look at the competitive exclusion principle from ecology, which says in its simplest form that two species competing for one resource cannot coexist, and show how the random switching can facilitate coexistence.

SEPC 2021 in honor of Elizabeth Meckes. Slides from the talks and more information are available <a href="https://services.math.duke.edu/~rtd/SEPC2021/SEPC2021.html">here</a>

Talk given at SEPC 2021 in honor of Elizabeth Meckes

Clouds and precipitation are among the most challenging aspects of weather and climate prediction. Moreover, our mathematical and physical understanding of clouds is far behind our understanding of a "dry" atmospheric where water vapor is neglected. In this talk, in working toward overcoming these challenges, we present new results on clouds and precipitation from two perspectives: first, in terms of the partial differential equations (PDEs) for atmospheric fluid dynamics, and second, in terms of stochastic models. A new asymptotic limit will be described, and it leads to new PDEs for a precipitating version of the quasi-geostrophic equations, now including phase changes of water. Also, a new energy will be presented for an atmosphere with phase changes, and it provides a generalization of the quadratic energy of a "dry" atmosphere. Finally, it will be shown that the statistics of clouds and precipitation can be described by stochastic differential equations and stochastic PDEs. As one application, it will be shown that, under global warming, the most significant change in precipitation statistics is seen in the largest events -- which become even larger and more probable -- and the distribution of event sizes conforms to the stochastic models.

The Langlands program is a far-reaching collection of conjectures that relate different areas of mathematics including number theory and representation theory. A fundamental problem on the representation theory side of the Langlands program is the construction of all (irreducible, smooth, complex) representations of certain matrix groups, called p-adic groups. In my talk I will introduce p-adic groups and provide an overview of our understanding of their representations, with an emphasis on recent progress. I will also briefly discuss applications to other areas, e.g. to automorphic forms and the global Langlands program.

I will discuss the influence of the seed in models of randomly growing trees; in particular, I will focus on the preferential attachment and uniform attachment models. In both of these models, different seeds lead to different distributions of limiting trees from a total variation point of view. I will discuss the differences and similarities in proving this for the two models. This is based on joint work with Sebastien Bubeck, Ronen Eldan, and Elchanan Mossel.

In this talk I will discuss joint work with J. Achter and C. Vial showing that the image of the Abel--Jacobi map on algebraically trivial cycles descends to the field of definition for smooth projective varieties defined over subfields of the complex numbers. The main focus will be on applications to topics such as: descending cohomology geometrically, a conjecture of Orlov regarding the derived category and Hodge theory, and motivated admissible normal functions.

Theta functions are automorphic forms on the double cover of symplectic groups and are important for constructing automorphic liftings. For higher-degree covers of symplectic groups, there are generalized theta representations and it is natural to ask if these ``higher'' theta functions play a similar role in the theory of metaplectic forms. In this talk, I will discuss new lifting of automorphic representations on the 4-fold cover of symplectic groups using such theta functions. A key feature is that this lift produces counterexamples of the generalized Ramanujan conjecture, which motivates a connection to the emerging ``Langlands program for covering groups'' by way of Arthur parameters. The crucial fact allowing this lift to work is that theta functions for the 4-fold cover still have few non-vanishing Fourier coefficients, which fails for higher-degree covers.

NOTE SEMINAR TIME: NOON!! Abstract: Let p be a prime number, F a field containing a primitive pth root of unity, and E/F a cyclic extension of degree p, with Galois group G. Let G_E be the absolute Galois group of E. The cohomology groups H^i(E,Fp)=Hî(G_E,Fp) possess a natural structure as FpG-modules and decompose into direct sums of indecomposables. In the 1960s Boreviè and Faddeev gave decompositions of E^*/E^*p -- the case i=1 -- for local fields. We describe the case i=1 for arbitrary fields, and then, using the Bloch-Kato Conjecture, we also determine the case i>1. No small surprise arises from the fact that there exist indecomposable FpG-modules which never appear in these module decompositions. We give several consequences of these results, notably a generalization of the Schreier formula for G_E, connections with Demu¹kin groups, and new families of pro-p-groups that cannot be realized as absolute Galois groups. These results have been obtained in collaboration with D. Benson, J. Labute, N. Lemire, and J. Mináè.

I will present a tutorial on the mathematical models utilized in molecular biology. I will begin with an introduction to the usual stochastic and deterministic models, and then introduce terminology and results from chemical reaction network theory. I will end by presenting the Â?deficiency zeroÂ? theorem in both the deterministic and stochastic settings.

We will give an explicit construction of the regular quotient of Morrissey-Ngô in the case of a symmetric pair. In the case of a quasisplit form (i.e. the regular centralizer group scheme is abelian), we will give a Galois description of the regular centralizer group scheme using parabolic covers. We will then describe how the nonseparated structure of the regular quotient recovers the spectral description of Hitchin fibers given by Schapostnik for U(n,n) Higgs bundles. This work is joint with B. Morrissey.

The large deviations analysis of solutions to stochastic differential equations and related processes is often based on approximation. The construction and justification of the approximations can be onerous, especially in the case where the process state is infinite dimensional. In this work we show how such approximations can be avoided for a variety of infinite dimensional models driven by some form of Brownian noise. The approach is based on a variational representation for functionals of Brownian motion. Proofs of large deviations properties are reduced to demonstrating basic qualitative properties (existence, uniqueness, and tightness) of certain perturbations of the original process. This is a joint work with P.Dupuis and V.Maroulas.

The Boltzmann equation governs the dynamics of a dilute gas. In this talk, I will address the L^p-stability problem of the Boltzmann equation near vacuum and a global Maxwellian. In a close-to-vacuum regime, I will explain the nonlinear functional approach motivated by Glimm's theory in hyperbolic conservation laws. This functional approach yields the uniform L^1-stability estimate. In contrast, in a close-to-global maxwellian regime, I will present the L^2-stability theory which establishes the uniform L^2-stability of several classical solutions.

The instanton equations appear in gauge theory and generalize both the Maxwell equations and the harmonic equation. Their study has been and continues to be a very fertile ground for interactions between physicists and mathematicians. The object of this talk is a description of instanton solutions on S^1xR^3 due to Hurtubise and myself using the Nahm transform, a non-linear transformation that takes a system of PDE and produces a system of ODE or even a system of algebraic equations. This description allows us to answer existence questions for calorons.

An SU(p, q)-flag domain is an open orbit of the real Lie group SU(p, q) acting on the complex flag manifold associated to its complexification SL(p + q, C). Any such flag domain contains certain compact complex submanifolds, called cycles, which encode much of the topological, complex geometric and repre- sentation theoretical properties of the flag domain. This talk is concerned with the description of these cycles in homology using a specific type of Schubert varieties. They are defined by the condition that the fixed point of the Borel group in question is in the closed SU(p,q)-orbit in the ambient manifold. We consider the Schubert varieties of this type which are of com- plementary dimension to the cycles. It is known that if such a variety has non-empty intersection with a certain base cycle, then it does so transversally (in finitely many points). With the goal of understanding this duality, we describe these points of intersection in terms of flags as well as in terms of fixed points of a given maximal torus. The relevant Schubert varieties are described in terms of Weyl group elements.

For Markov random fields temporal mixing, the time it takes for the Glauber dynamics to approach it's stationary distribution, is closely related to phase transitions in the spatial mixing properties of the measure such as uniqueness and the reconstruction problem. Such questions connect ideas from probability, statistical physics and theoretical computer science. I will survey some recent progress in understanding the mixing time of the Glauber dynamics as well as related results on spatial mixing. Partially based on joint work with Elchanan Mossel

This talk will be about the convergence problem in mean field control (MFC), i.e. the challenge of rigorously justifying the convergence of certain "symmetric" $N$-particle stochastic control problems towards their mean field counterparts. On the one hand, this convergence problem is already well-understood from a qualitative perspective, thanks to powerful probabilistic techniques based on compactness. On the other hand, quantitative results (i.e. rates of convergence) are more difficult to obtain, in large part because the value function of the mean field problem (which is also the solution to a certain Hamilton-Jacobi equation on the Wasserstein space) may fail to be $C^1$, even if all the data is smooth. After giving an overview of the convergence problem, I will discuss the results of two recent joint works with Cardaliaguet, Daudin, Delarue, and Souganidis, in which we use some ideas from the theory of viscosity solutions to overcome this lack of regularity and obtain rates of convergence of the $N$-particle value functions towards the value function of the corresponding MFC problem.

Using Calabi's method, I will construct rotationally symmetric Kahler- Ricci solitons on the total space of direct sum of fixed hermitian line bundle and its projective compactification, where the curvature of hermitian line bundle is Kahler-Einstein. These examples generalize the construction of Koiso, Cao and Feldman-Ilmanen-Knopf.

Recent studies of cholesterol-rich membrane microdomains, called caveolae, reveal that caveolae are reservoirs of recruitable sodium ion channels. Caveolar channels constitute a substantial and previously unrecognized source of sodium current in cardiac cells. In this talk, I will present a family of DE and PDE models to investigate caveolar sodium currents and their contributions to cardiac action potential morphology. We show that the b-agonist-induced opening of caveolae may have substantial impacts on peak overshoot, maximum upstroke velocity, and conduction velocity. Additionally, we show that prolonged action potentials and the formation of potentially arrhythmogenic afterdepolarizations, can arise if caveolae open intermittently throughout the action potential.

Motivated by the need to formulate and solve Plateau type problems in higher dimensions and codimensions, normal and integral currents were introduced by Federer and Fleming around 1960; their work was, to some extent a generalization of earlier work by DeGeorgi in codimension one as well as the work of Reifenberg in arbitrary codimensions. Since then a great deal of work has been done on the Plateau problem and related variational problems. This work has always been based on geometric measure theory. The so-called closure theorem for integral currents and the boundary rectifiability theorem are essential ingredients in all of this work; these theorems depend on the Besicovitch-Federer structure theory for set of finite Hausdorff measure in Euclidean space. More recently, in the work of Ambrosio and others, a useful theory of Sobolev spaces for functions with values in an arbitrary metric space has been developed and applied to a variety of problems. Ambrosio and Kirchheim have developed a theory of currents in metric spaces in which they are able to give geometrically appealing proofs of generalizations of the aforementioned closure and rectifiability theorems using some ideas of Almgren and DeGiorgi and avoiding the use of the Besicovitch-Federer structure theory. In this talk I will describe how they do it.

We consider the spectrum of the Dirichlet-Neumann map. This is the spectrum of the operator which sends a function on the boundary of a domain to the normal derivative of its harmonic extension. Along with the Dirichlet and Neumann spectrum, this problem has been much studied. We show how the problem of finding domains with fixed boundary area and largest first eigenvalue is connected to the study of minimal surfaces in the ball which meet the boundary orthogonally (free boundary solutions). We describe some conjectures on optimal surfaces and some progress toward their resolution. This is joint work with Ailana Fraser.

In this talk, I will present an overview of the framework originally proposed by Harder and further developed in collaboration with Raghuram to address rationality problems for special values of certain automorphic L-functions. I will then proceed to state my main results on the rationality of the special values of Langlands-Shahidi L-functions appearing in the constant term of the Eisenstein series associated with the exceptional group of type G_2â?? over a totally imaginary number field. This study marks the first instance where rank-one Eisenstein cohomology is employed to investigate the arithmetic of automorphic L-functions in the presence of multiple L-functions.

The principle of functoriality is a central question in present day mathematics. It is a far reaching, but quite precise, conjecture of Langlands that relates fundamental arithmetic information with equally fundamental analytic information. The arithmetic information arises from the solutions of algebraic equations. It includes data that classify algebraic number fields, and more general algebraic varieties. The analytic information arises from spectra of differential equations and group representations. It includes data that classify irreducible representations of reductive groups. The lecture will be a general introduction to these things. If time permits, we shall also describe recent progress that is being made on the problem.

One of the classical examples of hydrodynamic instability occurs when two fluids are separated by a free surface across which the tangential velocity has a jump discontinuity. This is called Kelvin-Helmholtz Instability. Kelvin-Helmholtz instability is a fundamental instability of incompressible fluid flow at high Reynolds number. The idealization of a shear layered flow as a vortex sheet separating two regions of potential flow has often been used as a model to study mixing properties, boundary layers and coherent structures of fluids. In a joint work with G. Hu and P. Zhang, we study the singularity of 3-D vortex sheets using a new approach. First, we derive a leading order approximation to the boundary integral equation governing the 3-D vortex sheet. This leading order equation captures the most singular contribution of the integral equation. Moreover, after applying a transformation to the physical variables, we found that this leading order 3-D vortex sheet equation de-generates into a two-dimensional vortex sheet equation in the direction of the tangential velocity jump. This rather surprising result confirms that the tangential velocity jump is the physical driving force of the vortex sheet singularities. It also shows that the singularity type of the three-dimensional problem is similar to that of the two-dimensional problem. Detailed numerical study will be provided to support the analytical results, and to reveal the generic form and the three-dimensional nature of the vortex sheet singularity.

After the sun, the oceans are the most significant contributor to our climate. Oceanic surface gravity waves are thought to have no influence on the global circulation of the oceans. However, oceanic surface gravity waves have a mean Lagrangian motion, the Stokes drift. This talk will present preliminary results that suggest that the dynamics of basin-scale oceanic currents are modified by the presence of the Stokes drift. In places where the Stokes drift is significant, it is possible that the ocean circulation, and hence climate, is not entirely well captured by present day models of the general circulation.

Special day: Thursday, 4pm, Room 120 Physics

Ensembles of redistricting plans can be challenging to analyze and visualize because every plan is an unordered set of shapes, and therefore non-Euclidean in at least two ways. I will describe two methods designed to address this challenge: barycenters for partitioned datasets, and a novel dimension reduction technique based on Gromov-Wasserstein distance. I will cover some of the theory behind these methods and show how they can help us untangle redistricting ensembles to find underlying trends. This is joint work with Ranthony A. Clark and Tom Needham.